Y. Sumino (Tohoku Univ.)

Y. Sumino(Tohoku Univ.)

Modern View of Perturbative QCD and Application to Heavy Quarkonium System

（現在の視点から見る摂動ＱＣＤ及び重いクォーコニウム系への応用）

☆Plan of Talk

1. Review of Pert. QCD (Round 1, Quick overview) • What’s Pert. QCD? • Today’s computational technologies

2. Review of Pert. QCD (Round 2, Some details) 3. Application to Heavy Quarkonium System

(4. More details of specific interests, upon request)

• physics in the heavy quark mass and interquark force

1. Review of Pert. QCD (Round 1, Quick overview)

What’s Pert. QCD?3 types of so-called “pert. QCD predictions” :(Confusing without properly distinguishing between them.)

(i) Predict observable in series expansion in

IR safe obs., intrinsic uncertainties

(ii) Predict observable in the framework of Wilsonian EFT

OPE as expansion in , uncertainties of (i) replaced by non-pert. matrix elements

(iii) Predict observable assisted by model predictions

Many obs in high-energy experiments depend on hadronization models, PDFs.Necessary (in MC) to compare with experimental dataSystematic uncertainties difficult to control, O(10%) accuracy at LHC

Do not add these non-pert. corr. to (i).

Remarkable progress of computational technologies in the last 10-20 years

(i) Higher-loop corrections

Resolution of singularities in multi-loop integrals Numerical and analytical methods Cross-over with frontiers of mathematics

(ii) Lower-order (NLO/NLL) corrections to complicated processes

Cope with proliferation of diagrams and many kinematical variablesMotivated by LHC physics

(iii) Factorization of scales in loop corrections

Provide powerful and precise foundation for constructing Wilsonian EFT

Dim. reg.: common theoretical basisEssentially analytic continuation of loop integralsContrasting/complementary to cut-off reg.

Comment on Impacts on Physics Insights:


cannot appear in series expansion in ?

To date, scattered over specialized fields, yet to frame a general overview

Examples:

• Various EFTs triggered new paradigms, such as HQET for b-physics, SCET for jets

new interpretations, viewpoints, concepts, …

2. Review of Pert. QCD (Round 2, Some details)




3 types of so-called “pert. QCD predictions” :











𝜇

ℒ𝑄𝐶𝐷 (𝛼𝑠 ,𝑚𝑖 ;𝜇)Theory of quarks and gluons

Predictable observables

𝑅 ( 𝐸 )≡𝜎 ¿¿• -ratio:

(ii) Observables of heavy quarkonium states (the only individual hadronic states)

(i) Inclusive observables (hadronic inclusive) insensitive to hadronization

• Inclusive decay widths

• Distributions of non-colored particles,

Same input parameters as full QCD.Systematic: has its own way of estimating errors.(Dependence on is used to estimate errors.)

Pert. QCD

• spectrum, leptonic decay width, transition rates

testable hypothesis

renormalization scale

Differs from a model

𝑞

𝑞

𝑞

𝑞

𝑞

𝑞

𝑞

𝑞

𝑅 (𝐸 )≡𝜎 ¿¿-ratio:

Renormalon uncertainty

𝛼𝑠(𝜇 )

𝛼𝑠 (𝜇)×𝑏0𝛼𝑠 (𝜇 ) log (𝜇𝑘 )

𝛼𝑠(𝜇 )×𝑏02𝛼𝑠2 (𝜇 ) log2(𝜇𝑘 )

𝑘

𝑘

𝑘

IR sensitivity at higher-order

𝑞

𝑞

𝑞

𝑞

𝑞

𝑞

𝛼𝑠(𝜇 )



𝑘

𝑘

𝑘

𝑘Λ

𝑞

𝑞

𝑞

𝑞

𝑞

𝑞

𝛼𝑠(𝜇 )



𝑘

𝑘

𝑘

Infinite sum

𝑞

𝑞

𝑞

𝑞

𝑞

𝑞

𝑘

𝑘

𝑘

Renormalon uncertainty

𝑘Λ

Consequence

𝑐𝑛 ( 𝐸 /𝜇 )𝛼𝑠𝑛 (𝜇)

~

Asymptotic series(Empirically good estimate of true corr.)Limited accuracy






Wilsonian EFTin terms of light quarks and IR gluons

1. Matching

2. Asymptotic expansion of diagrams

𝜇

𝐸integrate

out

ℒ𝑄𝐶𝐷

Determine Wilson coeffs such that physics at is unchanged,

via pert. QCD:

less d.o.f.

(𝜇 )=𝑔𝑖 𝑖𝒪∑𝑖 (𝑞𝑛 ,𝑞𝑛 ,𝐺𝜇)ℒEFT (𝜇 )

include only UV contr. Free from IR renormalon uncertainties

OPE in Wilsonian EFT multipole expansion

𝑘/𝑃≪1

𝜇

𝐸

Observable which includes a high scale

integrateout

light quarks and IR gluons

replace renormalons

















Dim. reg.

Advantages

• Preserves important symmetries (Lorentz sym, gauge sym)• In a single step, all loop integrals are rendered finite; both UV and IR. (cf. Pauli-Villars reg.)• Many useful computational techniques

Disadvantages

• Not defined as a quantum field theory (cf. lattice reg.) Nevertheless, well-defined and uniquely defined in pert. computations.• Difficult to interpret physically

Does represent IR or UV divergence? Unphysical equalities? Is only UV part of the theory modified?

(I can give an argument why I believe dim. reg. leads to correct predictions.)

Most powerful application of Dim. Reg.

Integration-by-parts (IBP) Identities Chetyrkin, Tkachov

0=∫𝑑𝐷𝑝𝑑𝐷𝑘 𝜕𝜕𝑘𝜇

𝑘𝜇

𝑝2𝑘2 (𝑘+𝑝 )2 (𝑝+𝑞 )2 (𝑘+𝑝+𝑞 )2

¿∫𝑑𝐷𝑝𝑑𝐷𝑘 1𝑝2𝑘2 (𝑘+𝑝 )2 (𝑝+𝑞 )2 (𝑘+𝑝+𝑞)2 [𝐷− 2𝑘 ∙𝑘𝑘2

−2𝑘 ∙ (𝑘+𝑝 )

(𝑘+𝑝 )2−2𝑘 ∙(𝑘+𝑝+𝑞)

(𝑘+𝑝+𝑞 )2 ]¿∫𝑑𝐷𝑝𝑑𝐷𝑘 1

𝑝2𝑘2 (𝑘+𝑝 )2 (𝑝+𝑞)2 (𝑘+𝑝+𝑞)2 [𝐷−4+ 𝑝2−𝑘2

(𝑘+𝑝 )2+

(𝑝+𝑞 )2−𝑘2

(𝑘+𝑝+𝑞 )2 ]

𝑘𝑝 𝑘+𝑝

𝑘+𝑝+𝑞

𝑞 𝑞

Standard technology used to reduce a large number of loop integrals to a small set of integrals (master integrals).

𝑝+𝑞

;

Example:

Most powerful application of Dim. Reg.

Integration-by-parts (IBP) Identities Chetyrkin, Tkachov

0=∫𝑑𝐷𝑝𝑑𝐷𝑘 𝜕𝜕𝑘𝜇

𝑘𝜇

𝑝2𝑘2 (𝑘+𝑝 )2 (𝑝+𝑞 )2 (𝑘+𝑝+𝑞 )2

¿∫𝑑𝐷𝑝𝑑𝐷𝑘 1𝑝2𝑘2 (𝑘+𝑝 )2 (𝑝+𝑞 )2 (𝑘+𝑝+𝑞)2 [𝐷− 2𝑘 ∙𝑘𝑘2

−2𝑘 ∙ (𝑘+𝑝 )

(𝑘+𝑝 )2−2𝑘 ∙(𝑘+𝑝+𝑞)

(𝑘+𝑝+𝑞 )2 ]¿∫𝑑𝐷𝑝𝑑𝐷𝑘 1

𝑝2𝑘2 (𝑘+𝑝 )2 (𝑝+𝑞)2 (𝑘+𝑝+𝑞)2 [𝐷−4+ 𝑝2−𝑘2

(𝑘+𝑝 )2+

(𝑝+𝑞 )2−𝑘2

(𝑘+𝑝+𝑞 )2 ]

𝑘𝑝 𝑘+𝑝

𝑘+𝑝+𝑞

𝑞 𝑞

Standard technology used to reduce a large number of loop integrals to a small set of integrals (master integrals).

𝑝+𝑞

;

Example:



Resolution of singularities in higher-loop integrals cross-over with frontiers of mathematics

(ii) Lower-order (NLO/NNLO/NLL) corrections to complicated processes

Cope with proliferation of diagrams and many variablesStrongly motivated by LHC physics



Dim. reg. as the common theoretical basis to all of themEssentially analytic continuation of loop integralsContrasting to cut-off reg.

Asymptotic Expansion of Diagrams

Simplified example:

(¿𝑀 )

𝑘

𝑝−𝑘

𝑞

𝑝−𝑞

𝑝𝑝𝑘−𝑞 ¿∫𝑑𝐷𝑘𝑑𝐷𝑞 1

𝑘2 (𝑝−𝑘)2 [ (𝑘−𝑞)2+𝑀 2 ]𝑞2 (𝑝−𝑞)2

in the case

Asymptotic expansion of a diagram and Wilson coeffs in EFT

Asymptotic expansion of a diagram and Wilson coeffs in EFT

𝑘

𝑝−𝑘

𝑞

𝑝−𝑞

𝑝𝑝𝑘−𝑞 ¿∫𝑑𝐷𝑘𝑑𝐷𝑞 1

𝑘2 (𝑝−𝑘)2 [ (𝑘−𝑞)2+𝑀 2 ]𝑞2 (𝑝−𝑞)2

L

L

L

L

LLL

= ¿1

𝑀 2

H

H

L

L

LLH

= ∫ 𝑑𝐷𝑘𝑘4 [𝑘2+𝑀 2 ]

H

H

H

H

LLH

= ∫ 𝑑𝐷𝑘𝑑𝐷𝑞𝑘4 [(𝑘−𝑞)2+𝑀 2 ]𝑞4= =

in the case

𝑝 ,𝑘 ,𝑞≪𝑀 𝑝 ,𝑞≪𝑘 ,𝑀 𝑝≪𝑘 ,𝑞 ,𝑀

Vertices and Wilson coeffs in EFT



Resolution of singularities in higher-loop integrals cross-over with frontiers of mathematics

(ii) Lower-order (NLO/NNLO/NLL) corrections to complicated processes

Cope with proliferation of diagrams and many variablesStrongly motivated by LHC physics

(iii) Factorizing and separating scales in loop corrections

Provide solid and precise foundation for constructing Wilsonian EFT


Theory of Multiple Zeta Values (MZV)

terms omitted

Example: Anomalous magnetic moment of electron ()

𝜁 (𝑛)=∑𝑚=1

∞ 1𝑚𝑛 2=−∑

𝑚=1

∞ (−1 )𝑚

𝑚ln ( 12 )= ∑

𝑚>𝑛> 0

∞ (−1 )𝑚+𝑛

𝑚3𝑛Li4

☆ Generalized Multiple Zeta Value (MZV)

Given as a nested sum

𝜁 (𝑛)=∑𝑚=1

∞ 1𝑚𝑛 2=−∑

𝑚=1

∞ (−1 )𝑚

𝑚ln ( 12 )= ∑

𝑚>𝑛> 0

∞ (−1 )𝑚+𝑛

𝑚3𝑛Li4

Can also be written in a nested integral form

∫0

1 𝑑𝑥𝑥 ∫

0

𝑥 𝑑𝑦𝑦−𝛼∫

0

𝑦 𝑑𝑧𝑧− 𝛽=−𝑍 (∞ ;2,1¿¿ ;)¿¿

e.g.1𝛼 ,

𝛼𝛽

MZVs can be expressed by a small set of basis (vector space over )

∑𝑚>𝑛> 0

∞ 1𝑚2𝑛

=∑𝑚=1

∞ 1𝑚3=𝜁 (3 )e.g. Dimension=1 at weight 3: .

For :

Shuffle relations are powerful in reducing MZVs. (Probably sufficient for .)

weight

MZV as a period of cohomology, motives

New relations for : Anzai,YS

#(MZVs)dim

weight

Relation between topology of a Feynman diagram and MZVs?

𝑍 (∞ ;3,1;𝑒𝑖 𝜋 /3 ,1 )= ∑𝑚>𝑛>0

∞ 𝑒𝑖𝑚 𝜋/3

𝑚3𝑛

𝑚=1

What kind of MZVs are contained in a diagram? Which s ?

Singularities in Feynman Diagrams

☆ Classes of singularities in a Feynman diagram

• IR singularity • UV singularity • Mass singularity • Threshold singularity

Complex -plane

cuts0+2 𝑖

−2 𝑖also log singularity at

𝑝

𝑞

𝑝+𝑞

𝑞

𝐼 (𝑞)≡∫𝑑4 𝑝 1(𝑝2+1 )2[ (𝑝+𝑞 )2+1 ]

∑𝑖∈𝐼

Singularities map

∫𝑑4𝑞 1(𝑞2+1 )2

𝐼 (𝑞)

𝑞 𝑞


𝑚=1

𝑚=1

𝑚=1

In simple cases all square-roots can be eliminated by (successive) Euler transf. Integrals convertible to MZVs

¿

Higher-order computationsIR renormalons increase of at IR

Pert. QCD

Summary of Overview

Higher-order computationsIR renormalons

Separation of UV & IR contr.Wilson coeffs vs. non-pert. matrix elements

OPE in Wilsonian EFTPert. QCD

Summary of Overview




replaced

only UV

Summary of Overview



Asymptotic expansion integration by region


Summary of Overview

replaced

Dim. reg.scale separation using analyticity

only UV



Asymptotic expansion integration by regions


Summary of Overview

replaced

Dim. reg.

Reduction by IBP identitiesResolution of singularities

scale separation using analyticity

only UV



Asymptotic expansion integration by region


Summary of Overview

replaced

Dim. reg.

Reduction by IBP identitiesResolution of singularities

MZVs

tough intermediate comp.of a diagram

SingularitiesTopology

scale separation using analyticity

final results very simple

only UV

short-cut ?

0.6% accuracy

0.8% accuracy

2% accuracy

3% accuracy ( 0.06% at ILC)

Precisions

Pert. QCD: Today’s benchmarks

Universality

More than 10 digits!

3. Application to Heavy Quarkonium System


• IR renormalization of Wilson coeffs in EFT

Anzai, Kiyo, YS

3-loop pert. QCD vs. lattice comp.

Static QCD Potential

𝑛𝑓 =0

Consider (naively) a “short-distance expansion”

at

According to renormalon analysis in pert. QCD, constant and term contain uncertainties

if we express the quark pole mass ()

IR renormalon in is canceled in the total energy

by the MS mass ().

𝒄−𝟏𝒓

2𝑚𝑝𝑜𝑙𝑒=2𝑚(1+𝑐1𝛼𝑠+𝑐2𝛼𝑠2+𝑐3𝛼𝑠

3+⋯)

Drastic improvement of convergence of pert. series


if we express the quark pole mass () by the MS mass ().

at 𝒄−𝟏𝒓


3+⋯)



if we express the quark pole mass () by the MS mass ().

at 𝒄−𝟏𝒓


3+⋯)


Exact pert. potential up to 3 loops

N=0

N=0

N=3

N=3

𝑟 [GeV-1]

𝑟 [GeV-1]

𝐴𝜇 (𝑞) 𝑗𝜇 (−𝑞)𝑞 Couples to total charge as .

𝑗𝜇 (𝑥 )=𝛿𝜇 0𝛿3( �⃗�−𝑟 /2)

General feature of gauge theory

𝐴𝜇 (𝑞) 𝑗𝜇 (−𝑞)𝑞 Couples to total charge as .

𝑗𝜇 (𝑥 )=𝛿𝜇 0𝛿3( �⃗�−𝑟 /2)

General feature of gauge theory

at

singlet octet singlet

US gluonOPE of QCD potential in Potential-NRQCD EFT

Uncetainty in replaced by a non-local gluon condensate within pNRQCD

cancel against

𝒄−𝟏𝒓

IR contributions

What are UV contributions?

A ‘Coulomb+Linear potential’ is obtained by resummation of logs in pert. QCD: YS

UV contributions

IR contributions

at

UV contributions

×

A ‘Coulomb+Linear potential’ is obtained by resummation of logs in pert. QCD: YS

Expressed by param. of pert. QCD

Coefficient of linear potential (at short-dist.)

𝜎 𝐿𝐿=2𝜋 𝐶𝐹

𝛽0(Λ𝑀𝑆

❑ )2

In the LL case 2𝜋

𝛽0 log (𝑞Λ𝑀𝑆

)

Coulombic pot. with log corr. at short-dist.

Formulas for

Define via

then

To see nature of , define Wilson coeff.in Potential-NRQCD for as

It can be proven that

This shows that, in pert. QCD, the “Coulomb” and linear parts of are determined by UV contributions and are independent of the factorization scale .

𝜇 𝑓

𝑞𝒓−𝟏

accurately predictable

Proof of

Hence,

Since , along we can expand

These terms are canceled and remain.

,

Subtraction of IR contributions in as contour integral around .

𝐸𝑡𝑜𝑡 (𝑟 )≈2𝑚+𝑐𝑜𝑛𝑠𝑡+𝑉 𝐶 (𝑟 )+𝜎 𝑟+𝑂 (Λ3𝑟2)

Heavy quarkonium spectrum Energy eigenvalues of

𝑟1𝑆2𝑆3𝑆 from linear pot. (predictable part)

for Coulomb splitting for Coulomb splitting

Implications

c.f. Rigorous computation in potential-NRQCD up to NNNLO

Rapid growth of masses of excited states originates fromrapid growth of self-energies of Q & Q due to IR gluons.

Brambilla, Y.S., Vairo

𝑎𝑋

good convergence

𝐸 𝑋≈2𝑚𝑏❑+∫

0

𝑚𝑏❑

𝑑𝑞 𝑓 𝑋 (𝑞 )𝛼𝑠(𝑞)

Rapid growth of masses of excited states originates fromrapid growth of self-energies of Q & Q due to IR gluons.

Brambilla, Y.S., Vairo

Mass of a bottomonium state mainly consists of(i) MS masses of and (ii) Contr. to the self-energies of and from gluons with wave-length Resemble difference of (state-dependent)constituent quark masses and MS masses.

__

__

𝑎𝑋

(1) One should carefully examine, from which power of non-pert. contributions start, and to which extent pert. QCD is predictable. (as you approach from short-distance region)

Messages:

(2) IR renormalization of Wilson coeffs.

𝜇 𝑓

𝑞𝒓−𝟏

SpectroscopyBottomonium spectrum at NNNLO

𝑑3=0.95×𝑑3𝑙𝑎𝑟𝑔𝑒−𝛽0

()

fixed at minimal-sensitivity scale for each level

Kiyo, YS

• Highly sensitive to . Stability practically determined by • Dependence on is minor.• Minimal-sensitivity scales are generally larger than at NNLO.

Standard form of loop integrals

Express each diagram in terms of standard integrals

1 loop

2 loop

3 loop

Each can be represented by a lattice site in N-dim. space

NB: is negative, when representing a numerator.

Integration-by-parts (IBP) Identities

Integration-by-parts (IBP) Identities

In dim. reg.

e.g. at 1-loop:

Chetyrkin, Tkachov

Reduction to Master Integrals (a small set of simple integrals)

𝑎 𝑏

𝑐

𝑚=1

𝑚=0

Singularities at or

MZVs with singularities at

IR UV

∑𝑖

𝑚𝑖=4∑𝑖

𝑚𝑖=2Another example

𝑞 𝑞

In simple cases all square-roots can be eliminated by (successive) Eulertransf. Integrals convertible to MZVs

Cause, however, proliferation of s

Proof of

Hence,

Since , along we can expand

These terms are canceled and remain.

,

Summary of Overview



inclusive obs./heavy quarkonium obs. uncertainties by higher-order corrections


separation of UV & IR contr.OPE: uncertainties of (i) replaced by non-pert. matrix elements UV Wilson coeffs. (pert. QCD with IR renormalization)


Many obs in high-energy experiments depend on hadronization models, PDFs.Necessary (in MC) to compare with experimental dataSystematic uncertainties difficult to control, O(10%) accuracy at LHC

increase of at IR

Remarkable progress of computational technologies


Resolution of singularities in higher-loop integrals Theory of MZVs in mathematics


Active development motivated by LHC physics pragmatic but no general (systematic) formulations as yet


Provide powerful and precise foundation for constructing Wilsonian EFTmay lead to new interpretation as substitute for cut-off reg.

Dim. reg. as the common theoretical basisEssentially analytic continuation of loop integralsContrasting/complementary to cut-off reg.

e.g. IBP id.

𝐸 𝑋≈2𝑚𝑏❑+∫

0

𝑚𝑏❑

𝑑𝑞 𝑓 𝑋 (𝑞 ) 𝛼𝑠(𝑞)

good convergence


3+⋯)

Microscopic View

OPE in Wilsonian EFT multipole expansion

ー

ー

+

+

gluon

gluon wave-length

𝜇

𝐸

Observable which includes a high scale

integrateout

light quarks and IR gluons

replace renormalons


Relation between topology of a Feynman diagram and MZVs?

𝑍 (∞ ;3,1;𝑒𝑖 𝜋 /3 ,1 )= ∑𝑚>𝑛>0

∞ 𝑒𝑖𝑚 𝜋/3

𝑚3𝑛

𝑚=0𝑚=1

𝜁 (5 )=𝑍 (∞;5 ;1)


terms omitted

Example: Anomalous magnetic moment of electron ()

𝜁 (𝑛)=∑𝑚=1

∞ 1𝑚𝑛 2=−∑

𝑚=1

∞ (−1 )𝑚

𝑚ln ( 12 )= ∑

𝑚>𝑛> 0

∞ (−1 )𝑚+𝑛

𝑚3𝑛Li4

Documents

Y. Sumino (Tohoku Univ.)